Two resistances R and 2R are connected in parallel in an electric circuit. The thermal energy developed in R and 2R are in the ratio

Options

A. (a)1:2
B. (b)2:1
C. (c )1:4
D. (d)4:1

Answer

To solve the problem of finding the ratio of thermal energy developed in two resistances \( R \) and \( 2R \) connected in parallel, we can follow these steps:

### Step 1: Understand the Concept of Heat Dissipation
The thermal energy (or heat) developed in a resistor can be calculated using the formula:
\[ H = I^2 R T \]
where \( H \) is the heat developed, \( I \) is the current flowing through the resistor, \( R \) is the resistance, and \( T \) is the time for which the current flows.

### Step 2: Apply Ohm's Law
From Ohm's Law, we know that:
\[ V = I R \quad \Rightarrow \quad I = \frac{V}{R} \]
where \( V \) is the voltage across the resistor.

### Step 3: Determine the Current through Each Resistor
For the resistors \( R \) and \( 2R \) in parallel, the voltage across both resistors is the same. Let:
- \( I_1 \) be the current through resistor \( R \)
- \( I_2 \) be the current through resistor \( 2R \)

Using Ohm's law:
\[ I_1 = \frac{V}{R} \quad \text{and} \quad I_2 = \frac{V}{2R} \]

### Step 4: Calculate Heat Developed in Each Resistor
Now, we can calculate the heat developed in each resistor:
- For resistor \( R \):
\[ H_1 = I_1^2 R T = \left(\frac{V}{R}\right)^2 R T = \frac{V^2}{R^2} R T = \frac{V^2 T}{R} \]

- For resistor \( 2R \):
\[ H_2 = I_2^2 (2R) T = \left(\frac{V}{2R}\right)^2 (2R) T = \frac{V^2}{4R^2} (2R) T = \frac{V^2 T}{2R} \]

### Step 5: Find the Ratio of Heat Developed
Now, we can find the ratio of heat developed in \( R \) and \( 2R \):
\[ \frac{H_1}{H_2} = \frac{\frac{V^2 T}{R}}{\frac{V^2 T}{2R}} = \frac{2R}{R} = 2 \]
Thus, the ratio of thermal energy developed in \( R \) to that in \( 2R \) is:
\[ H_1 : H_2 = 2 : 1 \]

### Conclusion
The thermal energy developed in resistances \( R \) and \( 2R \) are in the ratio \( 2 : 1 \).

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